From chance to order

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The models that Wendelin Werner and his colleagues study can be represented as m
The models that Wendelin Werner and his colleagues study can be represented as map-like figures. They study for instance universal properties of such jagged lines. (Graphic: Jason P. Miller)
In Nymphenburg on Friday, the ETH probabilist Wendelin Werner was awarded the Heinz Gumin Prize, the highest-value mathematics prize in Germany. But what does this have to do with a journey, a public transport map, and human emotions?

A conversation with Wendelin Werner is like a journey - and the paths that this journey follows were drawn by chance. Werner’s main field of research is probability theory, a branch of mathematics that deals with randomness.

Among other things, he studies random paths, which are known as ‘random walks’ because their individual steps occur one after the other at random. Probabilists like Werner are especially interested in the fact that, although random events look rather chaotic on the small scale, they nevertheless often give rise to structures that humans can recognise on the large scale in the continuous world. For example, on the large scale, a random walk becomes the so-called Brownian motion, which was first described by the Scottish botanist Robert Brown in 1827.

This theme is closely linked to statistical physics, where one derives the macroscopic rules that describe the behaviour of gases, for example, from the chaotic and random movements of innumerable microscopic particles.

Describing properties of random structures

On Friday, Wendelin Werner received the Heinz Gumin Prize for Mathematics from the Carl Friedrich von Siemens Foundation for his ‘pioneering contributions to the mathematical justification of universal properties of Brownian motion with applications to central assumptions of statistical physics’. For the French professor of mathematics, who has been at ETH Zurich since 2013 and who in 2006 received the Fields Medal - the most important award for mathematicians under the age of 40 - the prize is a testimony to fundamental research.

For Werner, the priority is not to calculate the probabilities of individual events; rather, he studies the mathematical structures or general properties that emerge when random events are transferred from the microto the macro-dimension: ‘I’m interested in what happens when many events occur together and a number of independent inputs becomes infinitely large. What global continuous structures are then formed, and what are their properties’’

From emotions to abstraction

And then he says something that may come as a surprise: ’Some people consider mathematics, highly abstract as it is, to be the polar opposite of the world of emotion. However, for me and for many other mathematicians, quite the reverse is true: the mathematical world, with its abstract ideas and concepts, actually opens up with the help of intuition and an emotional aspect. The creativity of the individual mathematician is a prerequisite for mathematical discovery.’

To grasp a mathematical structure, he says, you need to consider it from different perspectives and be able to link up a variety of ideas and concepts in an individual, creative way: ‘Every mathematician has a different image of a mathematical concept - be it because of their childhood, their experiences or certain stimuli from the outside world. Structures develop a life of their own in our brains, because every mathematician is different even if everyone is using the same definitions.’

A pathway through a network map

In order to illustrate the pathway to discovering a new mathematical idea, Werner uses the example of a public transport network map. A map of this kind is an abstract way of describing the stations, connections and interchange points in a public transport network. However, it does not depict the individual route chosen by a person travelling on the network and who wants, for example, to travel from ETH’s main building to Zürich-Altstetten station. This choice is made by each passenger individually, often without looking at the network map; instead, they rely on their own experience and memories. Their decisions are also determined by various influences from the outside world, such as the weather. And so everyone quickly picks out a specific pathway in their head in a somewhat abstract way.

‘The process is very similar when it comes to mathematical thoughts,’ says Werner. ‘Here, too, you begin a journey and have to see how to best link things up. In order to reach the destination, a mathematician not only needs a certain intuitive, global idea of the ’network map? but also has to make a series of specific, accurate decisions.’

In fact, Werner is especially interested in events that can be represented in map-like diagrams or figures. In this regard, special angle-preserving distortions of the images (the so-called conformal mappings) play an important role and link his work to other branches of mathematics.

At the boundaries

One specific model that Werner studies can be heuristically described as follows: if immune and infected cells are present on a surface and it is touch-and-go which cells will get the upper hand, then the cells will be distributed over specific areas, like on a map, and borders will form between these ‘territories’. Each infected section is surrounded by a jagged border. Although this forms randomly, it is possible to derive mathematically some of its structural properties, such as its ‘fractal dimension’, which roughly describes how long and convoluted it is.

Because the field of mathematics that Werner studies describes global structures and universal properties, models of this kind appear in numerous examples from the natural sciences. ‘Physics and the empirical world are constant sources of inspiration for me,’ he says. ‘In this way, mathematics is always connected with both the external and the internal, emotional reality.’